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Chapter 14

Moments and torque

14.1

M

Summary of moments and torques

Description

n

Expression

FQ /O

= r Q/O × FQ

i=1 MS/O

Moment of force FQ about point O Moment of a set S of forces about point O Shift theorem for moment of a force Torque is the moment of a set S of forces whose resultant is 0.

MS/O = MS/P = TS =

MF

Qi /O

+ r O/P × FS

MS/O

14.2

Moment of a vector

Q

FQ

The moment of a vector results from the cross product of a position vector with a bound vector.a The moment of the vector FQ (bound to point Q) about point O is Q denoted MF /O and is defined in terms of r Q/O (Q's position vector from O) as

a

r Q/O

O

(1)

A bound vector is a vector that is bound to a point.

MF

Q /O

= r Q/O × FQ

14.2.1

Moment arm of a vector about a point

Q

The moment arm of the bound vector FQ about point O is the distance d between O and the line of actiona of FQ and measures the effectiveness of FQ at creating a moment about O. This distance can be calculated in various ways, e.g, using the unit vector u in the direction of FQ and/or the angle between u and r Q/O . d =

a

FQ

d

r Q/O

O

r

Q/O

sin() =

r

Q/O

×u

Q

=

r Q/O · r Q/O

-

(r Q/O · u)2

Q

MF

Q /O

= FQ d

The line of action of the bound vector F

is the line passing through Q and parallel to F .

14.2.2

Moment of a set of vectors

MS/O = MS/O =

n i=1

The moment of a set S of bound vectors FQ1 , ..., FQn about a point O is defined as the sum of the moments of each bound vector in equation (2). In view of Newton's law of action/reaction in Section 13.5.5, the moment of a system S's internal forces is 0 and MS/O can be written much more simply as solely the moment of external forces on S. 141

MF

Qi /O

(2) (3)

(13.7)

MS/O

external

14.2.3

Statics, dynamics, and the moment of a set of forces

The moment MS/O of all forces on a system S about an arbitrary point O is related to the time-derivative of S's angular momentum about O in N and other quantities. This relationships simplifies to static equilibrium in equation (4) when the system S is at rest (not moving) in N or S is considered massless.

O

MS/O

(statics)

=

0

(4)

14.2.4

Shift theorem for the moment of a set of vectors

MS/P , the moment of a set S of bound vectors about a point P , can be calculated in terms of: · MS/O , the moment of S about a point O · r O/P , O's position vector from P · FS , the resultant of S

P

r O/P

O

(5)

MS/P = MS/O + r O/P × FS

14.2.5

Torque of a set of vectors

Torque is the moment of a set S of vectors whose resultant is zero. TS = MS/O

where

FS = 0

and point O is any point

(6)

Since a couple is a set of vectors whose resultant (sum) is 0, a torque is the moment of a couple.1 A couple has a special property, namely, the moment of a couple about a point O is equal to the moment of the couple about any other point Q. As a result, a torque is not associated with a point. The following example highlights the difference between a torque and a moment. Consider the various sets S of forces in Figure ??, and fill in the following table by calculating FS , the resultant of S, and MS/O , MS/P , MS/Q , the moments of S about points O, P , and Q, respectively. Express your results in terms of the right-handed orthogonal unit vectors nx , ny , nz . S A B C D FS 10 ny MS/O 50 nz MS/P 0 MS/Q MS/O = MS/P = MS/Q ? Yes/No Yes/No Yes/No Yes/No Moment is a torque? Yes/No Yes/No Yes/No Yes/No

14.2.6

Torque on a rigid body (or reference frame)

Torques can be associated with reference frames when the vectors that comprise the torque have a special character. The notation TA is used to designate a torque of a couple whose vectors F1 , ..., Fn have lines of actions that pass through points Q1 , ..., Qn , respectively, each of which is fixed in a reference frame A. When Q1 , ..., Qn are also fixed in a second reference frame, e.g., B, then one designates the torque TA/B .

Torques are a special type of Moment, namely the moment of a couple. All Torques are Moments, but not all Moments are Torques. A useful analogy is all Toyotas are Motor-vehicles, but not all Motor-vehicles are Toyotas.

Copyright c 1992-2009 by Paul Mitiguy

1

142

Chapter 14:

Moments and torque

10 n

10 n

A

O

5m

B

P

3m

Q

O

5m

P

3m

Q

ny

10 n

10 n

C

O

6n 5m

nz

Q

6n

nx

10 n

D

O

5m 4n

P

3m

P

3m 6n

Q

Figure 14.1: Various sets of forces

14.3

Proof of shift theorem for the moment of a set of bound vectors

To establish the validity of equation (5), consider a set S of bound vectors F1 , ..., Fn , whose lines of action pass through points Qi (i = 1, ..., n), respectively. The moment of S about a point P is defined in terms of r Qi /P (Qi 's position vector from P ) as MS/P =

n i=1

(2)

r Qi /P × FQi

(7)

Qi 's position vector from P may be written as r Qi /P = r O/P + r Qi /O Substituting equation (8) into equation (7), distributing the summation, and rearrangement yields MS/P

n (7,8)

(8)

= =

r O/P × FQi +

n i=1

n i=1 n i=1

r Qi /O × FQi r Qi /O × FQi (9)

i=1

r O/P ×

FQ i +

The first summation in equation (9) is the definition of FS (the resultant of S). The second summation in equation (9) is the definition of MS/O (the moment of S about O). Combining these two facts produces equation (5), the shift theorem for the moment of a set of bound vectors, namely MS/P = r O/P × FS + MS/O

Copyright c 1992-2009 by Paul Mitiguy

143

Chapter 14:

Moments and torque

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